In the novel Bitterblue, by Kristin Cashore, one character has a watch which is divided into 15 hours, each of which has 50 minutes. The novel has a brief discussion of how to convert time on that watch to standard time, and I'd like to look at it in a little more detail.

Like a conventional watch, which shows 12 hours of 60 minutes, the watch in Bitterblue shows half a day. However, the number of periods, and therefore the lengths of the periods, that it shows are different. One day has 24 standard hours, but 30 watch hours. (I will refer to times and durations on the watch as w-hours, w-minutes, etc. for clarity.) Therefore, there are 4 hours in 5 w-hours, or 1 w-hour = 4/5 hour (or 48 minutes). With 60 minutes in an hour, there are 1440 minutes in a day, but since there are only 50 w-minutes in a w-hour, there are 1500 w-minutes in a day. This means that 24 minutes = 25 w-minutes, so minutes and w-minutes have a similar duration.

With these relationships, it's possible to convert a time on the watch to a conventional time. One way to do it is to convert w-hours to w-minutes, then convert the total w-minutes to conventional minutes, then convert the minutes back to hours and minutes. If the w-time is h:m, the formula to convert to minutes is (h×50 + m) × 24/25. Divide this number by 60 to get the current hour, and the remainder is the current minutes. In Bitterblue, the title character does an example of a similar computation with equivalent results.

After Bitterblue does the calculation, she remarks that "I, for one, would find it simpler to memorize which time signifies what." As a halfway step to memorizing lots of times, it's fairly easy to estimate the time from the w-time. We'll start with a rough estimate that's accurate to about 5 minutes and then tighten it up a bit. 4 hours = 5 w-hours, so 4:00 = 5:00 w-time, 8:00 = 10:00 w-time, and 12:00 = 15:00 w-time. The first step is to find the closest current hour to one of these three points. The second step is to observe that 1 w-hour is 4/5 hour, and that 3/4 is close to 4/5. We're doing some rounding here, but we're used to thinking in quarter hours and we can correct the rounding later if we need to. So we start at 5:00, 10:00, or 15:00, and we add or subtract enough w-hours to be close to the current w-time. For each w-hour added or subtracted, we add or subtract 3/4 hour from the time. The last step is to add or subtract w-minutes. Since we're just estimating, and 1 w-minute = 24/25 minute, we can just add or subtract the number of w-minutes after or before the hour and ignore the conversion.

Let's do an example. Say the w-time is 8:35. 8 is close to 10, so we start there. 10:00 w-time = 8:00. Then we subtract 1 w-hour from 10:00 to get 9:00, so we subtract 45 minutes from 8:00 to get 7:15. Finally, we subtract 15 w-minutes from 9:00 to get 8:35. (The minutes subtraction is the step which throws me. Since there are 50 w-minutes in a w-hour, 8:35 is 15 w-minutes before the hour, not the expected 25.) Subtracting 15 minutes from 7:15 gives our estimate of 7:00. 8:35 on the watch is approximately 7:00 normal time.

We did some rounding, which we can now correct if we need more precision. We approximated 1 w-hour as 3/4 hour, when it's really 4/5 hour. 3/4 is 45 minutes and 4/5 hour is 48 minutes, so we can add or subtract 3 additional minutes per w-hour. In this case, 6:57 is a closer estimate than 7:00. Finally, there's a small rounding error in the minutes. If we add close to 25 w-minutes, we should subtract 1 minute from our estimate, and vice versa. Since we subtracted 15 w-minutes, it's slightly closer to add 1 minute back in, for a final time of 6:58.

Doing the full computation, we get 8×50 + 35 = 435. 435×24 is 10440. 10440/25 is 417.6. 417.6 divided by 60 is 6, with a remainder of 57.6. In other words, the exact time is 6:57:36. I can do 435×24 in my head, but I don't really want to, and estimating that the current time is about 7:00 was much easier and probably accurate enough.

Often, when I look at a watch face, I don't actually need to know the exact time. I'm just looking for a quick estimate, based on the hand position. So what does the hand position on a 15 hour watch tell us about the standard time? On a standard watch, the hour hand travels a full circle in one half day, so the angle from vertical tells us the exact time. It's easy to judge that if the hour hand is pointing down and a little to the left, it's about 7:00, and based on the exact angle it's not hard to judge whether it's closer to 6:30, 7:00, or 7:30, without even referring to the minute hand.

On the 15 hour watch, the hour hand travels a full circle in one half day, exactly the same as a standard watch. So the angle of the hour hand is the same as on the standard watch. You can work out the angles from the example, and you will find that the angle of the hour hand at 8:35 on the 15 hour watch is nearly the same as the angle of the hour hand at 7:00 on a standard watch. This means that a quick glance at the hour hand on a 15 hour watch will give you exactly the same information as a quick glance at the hour hand on a standard watch.

The minute hand is a different story. The minute hand on a 15 hour watch completes one full circle in 48 (standard) minutes, which means that it points all kinds of different directions relative to the standard minute hand. In our example, at 8:35, the minute hand is pointing to the left and slightly down on the 15 hour watch, but at 7:00 on a standard watch it's pointing straight up. It's hard to get useful information out of the minute hand without doing the full time conversion, either estimated or using the exact formula.

One other point about the watch face design on the 15 hour watch: On a standard watch, it's possible for one set of marks to indicate both hours and minutes. The angle representing 1 hour on the hour hand is the same as the angle representing 5 minutes on the minute hand. So you can mark just the hours, and it's easy to read the minutes just off the hour markings. On the 15 hour watch, things don't work out so well. The angle representing 1 w-hour is the angle for 3 1/3 w-minutes. To read this watch with the same ease and precision as a standard watch, you really need two sets of marks, one for w-hours and one for w-minutes.